To determine which number would represent an outlier in the given dataset, we need to find the range of the existing values and then see which of the provided options falls outside the typical range of the data.
The dataset provided is: 42, 47, 50, 42, 45, 41, 49, 51, and 44.
First, let's calculate the mean and standard deviation of this dataset to help identify an outlier:
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Find the mean:
- Sum of all observations: 42 + 47 + 50 + 42 + 45 + 41 + 49 + 51 + 44 = 411
- Number of observations: 9
- Mean = 411 / 9 ≈ 45.67
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Find the standard deviation:
- Calculate the variance:
- Deviations from the mean:
- (42 - 45.67)² = 13.49
- (47 - 45.67)² = 1.78
- (50 - 45.67)² = 18.51
- (42 - 45.67)² = 13.49
- (45 - 45.67)² = 0.45
- (41 - 45.67)² = 21.78
- (49 - 45.67)² = 10.91
- (51 - 45.67)² = 28.51
- (44 - 45.67)² = 2.78
- Variance = (Sum of squared deviations) / (Number of observations)
- Variance ≈ (13.49 + 1.78 + 18.51 + 13.49 + 0.45 + 21.78 + 10.91 + 28.51 + 2.78) / 9 ≈ 10.66
- Standard deviation ≈ √10.66 ≈ 3.26
- Deviations from the mean:
- Calculate the variance:
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Identify outliers:
- Outliers are typically defined as any point that lies more than 1.5 times the interquartile range above the third quartile or below the first quartile, or any point that lies more than 2 standard deviations away from the mean.
- Mean = 45.67; 2 * Standard Deviation ≈ 2 * 3.26 ≈ 6.52
- Thus, any value beyond 45.67 + 6.52 ≈ 52.19 or below 45.67 - 6.52 ≈ 39.15 can be considered an outlier.
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Analyze options:
- 72: This is clearly above 52.19 and would definitely be an outlier.
- 40: This is below 39.15 and would also be an outlier.
- 52: This is slightly above the mean but not more than 2 standard deviations, so it’s not an outlier.
- 44: This is within the range of the data and not an outlier.
Given these analyses, both 72 and 40 would be considered outliers, but since the question asks for ONE number that would represent an outlier, 72 is the most distinctly outside the upper range of results.
So, the answer is 72.